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In a given solubility equilibrium, I know that if $ Q >K_s$, then the salt will precipitate until $ Q =K_s$. If $ Q <K_s$, however, would the precipitate dissolve until $ Q =K_s$ too?
You've described the general tendency for any equilibrium process, including solubility. So, the short answer is yes, you will dissolve more until the reaction quotient is equal to the equilibrium constant.
Required, but never shown